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Twisted foliated fracton order

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In the study of three-dimensional gapped models, two-dimensional gapped states can be considered as a free resource. This is the basic idea underlying our proposal of the notion of `foliated fracton order'. Using this idea, we have found that many of the known type-I fracton models, like the X-cube model and the checkerboard model, have the same foliated fracton order. In this talk, I will present three-dimensional fracton models with a different kind of foliated fracton order. The previously known foliated fracton order corresponds to the gauge theory of a simple paramagnet with subsystem planar symmetry. The new order corresponds to a twisted version of the gauge theory where the system before gauging has nontrivial order protected by the subsystem planar symmetries. I will discuss a way to identify the nontrivial order by compactifying the system in the z direction and analyzing the resulting two dimensional order.